The New York Times has been one of President Donald Trump's biggest targets as he rails against the media and what he calls "fake news," but the media company's CEO told CNBC on Thursday the paper aims to be objective in its coverage.

In fact, while Trump has described The New York Times as "failing," Mark Thompson said it is anything but.

"This is the danger with fake news. Even the president of the United States can be taken in by it and end up saying things which aren't true. We're not failing. We are growing our audiences. We are growing our subscriptions," Mark Thompson said in an interview with "Power Lunch."

Trump has tweeted several times about the The New York Times since winning the election. Earlier this week he advocated someone buy the "failing" newspaper.

Thompson said it was unclear how many of those new subscribers came to the newspaper because of Trump.

"It's Donald Trump, but also it's a bigger story about political controversy and division in this country and across the Western world. It's Brexit. It's an uncertain Europe. It's a world which feels both politically and economically very unstable," he noted.

As for those who have canceled their subscriptions, they all had different reasons — from being sick of the news to believing the Times was too negative about Trump or about Clinton, said Thompson.

"There are Hillary loyalists who were upset the Times broke the story about Hillary's private servers, for example. There was a feeling, amongst some of our readers, that we'd been too hard on Hillary."

'Naked and reckless falsehoods'

Meanwhile, the media have now found themselves in an era where what they are reporting is being questioned.

While being challenged is part of a free society, Thompson said there is a striking phenomenon occurring now where "naked and reckless falsehoods" are being spread by politicians around the world.

"There is a difference between striving for the truth, striving to understand, professional journalists, professional editors doing their best to understand and convey that to the public, which I honestly believe the New York Times newsroom tries to do, and people who are just literally making it up," he said.

Those made-up stories are then being spread by politicians, he added.

And Thompson said Trump is not immune.

While Trump has been "astonishingly open" with the media and has even answered his own phone, there is a flip side, Thompson said.

"On the other hand, manifestly, one of the points for the Times and other media has been repeatedly having to point out to the public that statements which have come from the president and the White House are not actually true," he said.